It's hard to say exactly as it depends on what the owning partner may have specified in a will or trust but generally:
The house would pass to the children.
The children would own the "benefit" but can't be registered as the legal owners until 18, so the house would need to be held in trust for them by a trustee who would be named the legal owner.
I think the fact of the trust would need to be registered as a notice or caution on the Land Register to stop the trustee disposing of the property without accounting for it to the children.
Who would the trustee be? Quite possibly the unmarried partner/children parent but depends on whether the deceased has specified someone else. If no trust already set up or specified the executor odlf the will/administrator of the estate will have to set up the terms of the trust and appoint trustees as he/ she sees fit.
Would the partner be able to remain living in the property with the children until they are both 18? Quite possibly and particularly likely if appointed trustee but the trust would need to specify how the mortgage will be paid and cost of repairing/maintaining/insuring the house is funded as the children have no income. Possibly this could be a deemed responsibility of the partner/trustee in lieu of paying rent to the children. (I'm speculating here to give you a rough idea of how it might work.) If the costs can't be met in this way then the house might need to be sold and proceeds placed in trust for the children and the surviving parent would need to make other arrangements for housing the children and themself. There will be limits on how much the children's money can be used towards that purpose.
If the house is not sold - Once both children are 18 the trust will be discharged and they will be the registered legal owners of the house and responsible for the upkeep and financing. In theory they could throw out the surviving parent but in reality that is unlikely if relations are good. But the surviving parent would have no financial interest in the property.